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Mishnah Berurah (93:5 – on seif 2)
The achronim write that whoever does not close his eyes when he pray the shemoneh esrei will not earn the privilege of seeing the countenance of the Shechinah (l’ros pnei hashechinah) when his soul departs. Notwithstanding, if one prays from a siddur and his eyes are open so that he can see what is written in it, there is no objection in doing so. {Says to be see the M’B 91:6 that brings a scary Zohar about looking around during shemoneh esrei] (Translation mostly from Feldheim)
I believe the Ari Z’l davened from a siddur so as to be careful to pronounce the words correctly.
In conclusion, both closing eyes and looking at a siddur are viable options, looking anywhere else is not during shemoneh esrei.